Optimal weighted centroid designs for maximal parameter subsystem for third degree Kronecker model mixture experiments

Abstract

Experiments that involve a mixture of ingredients are usually associated with investigating optimal proportions of several factors used. Optimal designs lower the costs of experimentation by allowing statistical models to be estimated with fewer experimental runs. Thus, appropriate designs for experiments that allow for parameter estimation without bias and with minimum variance are desirable. The purpose of this study was to obtain optimal weighted centroid designs for maximal parameter subsystem for third degree Kronecker model mixture experiments with the assumption that errors are independent and identically distributed with mean zero and common variance. The general objective was to obtain optimal weighted centroid designs for maximal parameter subsystems for third degree Kronecker model mixture experiments. The specific objectives of the study were to: Identify the coefficient matrix K and the associated parameter subsystem of interest; determine optimal moments and information matrix for two, three, four, and generalized to m factors; derive optimal weighted centroid designs for third degree Kronecker model for mixture experiments for A-, D- and E-optimality criteria and finally, compute numerical optimal weighted centroid designs for the maximal parameter subsystem. The Kronecker model approach was used to obtain the coefficient matrix K and, consequently, the optimal moments. A set of weighted centroid designs for the maximal parameter subsystem of interest was obtained by the use of unit vectors and characterization of feasible weighted centroid designs for the parameter subsystem. Information matrices based on maximal parameter subsystem were also obtained for the two, three, four, and generalized to m factors. Kiefer-Wolfowitz equivalence theorem was used to derive weights for the respective weighted centroid designs for D-, A- and E- Optimality. Optimal weights and values were computed numerically using Wxmaxima and R software. The results obtained indicated that: Coefficient matrix K obtained had a full column rank and helped in the identification of the linear parameter subsystem; the optimal moments obtained reflected the statistical properties of designs and were useful in finding the information matrix; the information matrix was important in obtaining ( p ) ( p ) optimality criteria and with  1 and  2 being the weights, for the average- variance criterion (A- criterion) and the optimality criteria were both dependent on ( p ) the information matrix, as the number of m factors increases,  1 decreases ( p ) while  2 increases and the value of the maximum criterion decreases. For the determinant criterion (D-criterion), as the number of m factors increase,  1 decreases while  increases and the value of the maximum criterion decreases. For the smallest eigenvalue criterion (E-criterion) as the number of m factors ( p ) ( p ) 2 increases,  1 increases while  2 decreases and the value of the maximum criterion decreases. This indicates that the maximal parameter design reflects well the statistical properties due to increasing symmetry as the number of factors increases. In conclusion, based on the maximal parameter subsystem third degree mixture model with two, three, four, and generalized to m factors for D-, A- and E- optimal weighted centroid designs for the parameter subsystem exists. The study thus recommends the application of the designs obtained by experimenters in designing of experiments to yield Optimal results in technological fields. This study concentrated on optimal weighted centroid designs for maximal parameter subsystem for third degree Kronecker model mixture experiments.